Influence of fast interstellar gas flow on the dynamics of dust grains

The orbital evolution of a dust particle under the action of a fast interstellar gas flow is investigated. The secular time derivatives of Keplerian orbital elements and the radial, transversal, and normal components of the gas flow velocity vector at the pericentre of the particle’s orbit are derived. The secular time derivatives of the semi-major axis, eccentricity, and of the radial, transversal, and normal components of the gas flow velocity vector at the pericentre of the particle’s orbit constitute a system of equations that determines the evolution of the particle’s orbit in space with respect to the gas flow velocity vector. This system of differential equations can be easily solved analytically. From the solution of the system we found the evolution of the Keplerian orbital elements in the special case when the orbital elements are determined with respect to a plane perpendicular to the gas flow velocity vector. Transformation of the Keplerian orbital elements determined for this special case into orbital elements determined with respect to an arbitrary oriented plane is presented. The orbital elements of the dust particle change periodically with a constant oscillation period or remain constant. Planar, perpendicular and stationary solutions are discussed. The applicability of this solution in the Solar System is also investigated. We consider icy particles with radii from 1 to 10 μm. The presented solution is valid for these particles in orbits with semi-major axes from 200 to 3000 AU and eccentricities smaller than 0.8, approximately. The oscillation periods for these orbits range from 10⁵ to 2 × 10⁶ years, approximately.     

Secular orbit variation due to solar radiation effects: a detailed model for BYORP

A detailed derivation of the effect of solar radiation pressure on the orbit of a body about a primary orbiting the Sun is given. The result is a set of secular equations that can be used for long-term predictions of changes in the orbit. Solar radiation pressure is modeled as a Fourier series in the body’s rotation state, where the coefficients are based on the shape and radiation properties of the body as parameters. In this work, the assumption is made that the body is in a synchronous orbit about the primary and rotates at a constant rate. This model is used to write explicit variational equations of the energy, eccentricity vector, and angular momentum vector for an orbiting body. Given that the effect of the solar radiation pressure and the orbit are periodic functions, they are readily averaged over an orbit. Furthermore, the equations can be averaged again over the orbit of the primary about the Sun to give secular equations for long-term prediction. This methodology is applied to both circular and elliptical orbits, and the full equations for secular changes to the orbit in both cases are presented. These results can be applied to natural systems, such as the binary asteroid system 1999 KW4, to predict their evolution due to the Binary YORP effect, or to artificial Earth orbiting, nadir-pointing satellites to enable more precise models for their orbital evolution.     

The 1/1 resonance in extrasolar planetary systems

We study orbits of planetary systems with two planets, for planar motion, at the 1/1 resonance. This means that the semimajor axes of the two planets are almost equal, but the eccentricities and the position of each planet on its orbit, at a certain epoch, take different values. We consider the general case of different planetary masses and, as a special case, we consider equal planetary masses. We start with the exact resonance, which we define as the 1/1 resonant periodic motion, in a rotating frame, and study the topology of the phase space and the long term evolution of the system in the vicinity of the exact resonance, by rotating the orbit of the outer planet, which implies that the resonance and the eccentricities are not affected, but the symmetry is destroyed. There exist, for each mass ratio of the planets, two families of symmetric periodic orbits, which differ in phase only. One is stable and the other is unstable. In the stable family the planetary orbits are in antialignment and in the unstable family the planetary orbits are in alignment. Along the stable resonant family there is a smooth transition from planetary orbits of the two planets, revolving around the Sun in eccentric orbits, to a close binary of the two planets, whose center of mass revolves around the Sun. Along the unstable family we start with a collinear Euler–Moulton central configuration solution and end to a planetary system where one planet has a circular orbit and the other a Keplerian rectilinear orbit, with unit eccentricity. It is conjectured that due to a migration process it could be possible to start with a 1/1 resonant periodic orbit of the planetary type and end up to a satellite-type orbit, or vice versa, moving along the stable family of periodic orbits.     

Secular and tidal evolution of circumbinary systems

We investigate the secular dynamics of three-body circumbinary systems under the effect of tides. We use the octupolar non-restricted approximation for the orbital interactions, general relativity corrections, the quadrupolar approximation for the spins, and the viscous linear model for tides. We derive the averaged equations of motion in a simplified vectorial formalism, which is suitable to model the long-term evolution of a wide variety of circumbinary systems in very eccentric and inclined orbits. In particular, this vectorial approach can be used to derive constraints for tidal migration, capture in Cassini states, and stellar spin–orbit misalignment. We show that circumbinary planets with initial arbitrary orbital inclination can become coplanar through a secular resonance between the precession of the orbit and the precession of the spin of one of the stars. We also show that circumbinary systems for which the pericenter of the inner orbit is initially in libration present chaotic motion for the spins and for the eccentricity of the outer orbit. Because our model is valid for the non-restricted problem, it can also be applied to any three-body hierarchical system such as star–planet–satellite systems and triple stellar systems.     

Poynting-Robertson drag and orbital resonance

Both the Poynting-Robertson drag and resonant orbits appear to be very important for the motion of small grains in the early solar system. While orbital resonances are very often stable and tend to force bodies into noncircular orbits, the Poynting-Robertson drag produces secular variations in the semimajor axis and tends to circularize the orbits. We study numerically the competition between the Poynting-Robertson drag and the gravitational interaction of grains with Jupiter near the 2/1 resonance. Computations are based on the plane-restricted problem. Numerical investigations show that the grains always cross the resonance region without any oscillation, except in the special case where the grains were initially inside the resonance. In both cases the variations of the osculating elements exhibit a drastic step, which can be explained by Greenberg's and Schubart's theories.     

Hysteresis effects and diagnostics of the shock formation in low angular momentum axisymmetric accretion in the Kerr metric

The secular evolution of the purely general relativistic low angular momentum accretion flow around a spinning black hole is shown to exhibit hysteresis effects. This confirms that a stationary shock is an integral part of such an accretion disc in the Kerr metric. The equations describing the space gradient of the dynamical flow velocity of the accreting matter have been shown to be equivalent to a first order autonomous dynamical systems. Fixed point analysis ensures that such flow must be multi-transonic for certain astrophysically relevant initial boundary conditions. Contrary to the existing consensus in the literature, the critical points and the sonic points are proved not to be isomorphic in general, they can form in a completely different length scales. Physically acceptable global transonic solutions must produce odd number of critical points. Homoclinic orbits for the flow possessing multiple critical points select the critical point with the higher entropy accretion rate, confirming that the entropy accretion rate is the degeneracy removing agent in the system. However, heteroclinic orbits are also observed for some special situation, where both the saddle type critical points of the flow configuration possesses identical entropy accretion rate. Topologies with heteroclinic orbits are thus the only allowed non-removable degenerate solutions for accretion flow with multiple critical points, and are shown to be structurally unstable. Depending on suitable initial boundary conditions, a homoclinic trajectory can be combined with a standard non-homoclinic orbit through an energy preserving Rankine-Hugoniot type of stationary shock, and multi-critical accretion flow then becomes truly multi-transonic. An effective Lyapunov index has been proposed to analytically confirm why certain class of transonic flow cannot accommodate shock solutions even if it produces multiple critical points.     

First order planetary perturbations with elliptic functions

The differential equations of planetary theory are solved analytically to first order for the two-dimensional case, using only Jacobian elliptic functions and the elliptic integrals of the first and second kind. This choice of functions leads to several new features potentially of importance for planetary theory. The first of these is that the solutions do not require the expansion of the reciprocal of the distance between two planets, even for those variables which depend on two angular arguments. A second result is that the solution is free from small divisors with the exception of two special resonances. In fact, not only are the solutions for resonant orbits free from small divisors, the perturbations for all variables are expressible in closed form. A subset of the resonant orbits maintains this form and in addition has the remarkable feature that the first order perturbations are purely periodic; they contain no secular terms. A solution for the 13 resonance case is given as an example.     

Resonant motion in the restricted three body problem

The resonant structure of the restricted three body problem for the Sun- Jupiter asteroid system in the plane is studied, both for a circular and an elliptic orbit of Jupiter. Three typical resonances are studied, the 2 : 1, 3 : 1 and 4 : 1 mean motion resonance of the asteroid with Jupiter. The structure of the phase space is topologically different in these cases. These are typical for all other resonances in the asteroid problem. In each case we start with the unperturbed two-body system Sun-asteroid and we study the continuation of the periodic orbits when the perturbation due to a circular orbit of Jupiter is introduced. Families of periodic orbits of the first and of the second kind are presented. The structure of the phase space on a surface of section is also given. Next, we study the families of periodic orbits of the asteroid in the elliptic restricted problem with the eccentricity of Jupiter as a parameter. These orbits bifurcate from the families of the circular problem. Finally, we compare the above families of periodic orbits with the corresponding families of fixed points of the averaged problem. Different averaged Hamiltonians are considered in each resonance and the range of validity of each model is discussed.     

Symmetric and asymmetric 3:1 resonant periodic orbits with an application to the 55Cnc extra-solar system

We study the dynamics of 3:1 resonant motion for planetary systems with two planets, based on the model of the general planar three body problem. The exact mean motion resonance corresponds to periodic motion (in a rotating frame) and the basic families of symmetric and asymmetric periodic orbits are computed. Four symmetric families bifurcate from the family of circular orbits of the two planets. Asymmetric families bifurcate from the symmetric families, at the critical points, where the stability character changes. There exist also asymmetric families that are independent of the above mentioned families. Bounded librations exist close to the stable periodic orbits. Therefore, such periodic orbits (symmetric or asymmetric) determine the possible stable configurations of a 3:1 resonant planetary system, even if the orbits of the two planets intersect. For the masses of the system 55Cnc most of the periodic orbits are unstable and they are associated with chaotic motion. There exist however stable symmetric and asymmetric orbits, corresponding to regular trajectories along which the critical angles librate. The 55Cnc extra-solar system is located in a stable domain of the phase space, centered at an asymmetric periodic orbit.     

Secular perturbations of asteroids in secular resonance

When the precessional rate of the orbital plane of an asteroid is nearly equal to that of Jupiter, the orbital inclination of the asteroid changes quite largely due to this near equality of their precessional rates, which is called a secular resonance. In the vicinity of the exact resonance the difference of their longitudes of nodes librates with quite a long period of order of 1×10⁶ yr. In this paper we treat this secular resonance by a method of semianalytical secular perturbations with use of numerical averaging for both non-resonant and resonant asteroids and show that the results by the semi-analytical treatment agrees qualitatively with those obtained by direct numerical integrations of asteroid's orbits.     

Orbits and manifolds near the equilibrium points around a rotating asteroid

We study the orbits and manifolds near the equilibrium points of a rotating asteroid. The linearised equations of motion relative to the equilibrium points in the gravitational field of a rotating asteroid, the characteristic equation and the stable conditions of the equilibrium points are derived and discussed. First, a new metric is presented to link the orbit and the geodesic of the smooth manifold. Then, using the eigenvalues of the characteristic equation, the equilibrium points are classified into 8 cases. A theorem is presented and proved to describe the structure of the submanifold as well as the stable and unstable behaviours of a massless test particle near the equilibrium points. The linearly stable, the non-resonant unstable, and the resonant equilibrium points are discussed. There are three families of periodic orbits and four families of quasi-periodic orbits near the linearly stable equilibrium point. For the non-resonant unstable equilibrium points, there are four relevant cases; for the periodic orbit and the quasi-periodic orbit, the structures of the submanifold and the subspace near the equilibrium points are studied for each case. For the resonant equilibrium points, the dimension of the resonant manifold is greater than 4, and we find at least one family of periodic orbits near the resonant equilibrium points. As an application of the theory developed here, we study relevant orbits for the asteroids 216 Kleopatra, 1620 Geographos, 4769 Castalia and 6489 Golevka.     

On the habitability of the OGLE-2006-BLG-109L planetary system

We investigate the dynamics of putative Earth-mass planets in the habitable zone (HZ) of the extrasolar planetary system OGLE-2006-BLG-109L, a close analogue of the Solar system. Our work is inspired by the work of Malhotra & Minton. Using the linear Laplace–Lagrange theory, they identified a strong secular resonance that may excite large eccentricity of orbits in the HZ. However, due to uncertain or unconstrained orbital parameters, the subsystem of Jupiters may be found in a dynamically active region of the phase space spanned by low-order mean-motion resonances. To generalize this secular model, we construct a semi-analytical averaging method in terms of the restricted problem. The secular orbits of large planets are approximated by numerically averaged osculating elements. They are used to calculate the mean orbits of terrestrial planets by means of a high-order analytic secular theory developed in our previous works. We found regions in the parameter space of the problem in which stable, quasi-circular orbits in the HZ are permitted. The excitation of eccentricity in the HZ strongly depends on the apsidal angle of jovian orbits. For some combinations of that angle, eccentricities and semimajor axes consistent with the observations, a terrestrial planet may survive in low eccentric orbits. We also study the effect of post-Newtonian gravity correction on the innermost secular resonance.     

On the stability of circular ‘asteroid’ orbits in anN-planetary system

The restricted problem of the motion of a point of negligible mass (asteroid) in anN-planetary system is considered. It is assumed that all the planets move about the central body (Sun) along circular orbits in the same plane and the mean motions of the asteroid and the planets are incommensurable. The asteroid orbit evolution is described as a first approximation by secular equations with the perturbing function averaged by the mean longitudes of the asteroid and the planets. For small values of the asteroid orbit eccentricity an expression for the secular part of the perturbing function has been obtained. This expression holds for the arbitrary values of the asteroid orbit semiaxis which are different from those of the planet orbit radii. The stability of the asteroid circular orbits in a linear approximation with respect to the eccentricity is studied. The critical inclinations for a Solar system model are calculated.     

The motion of charged dust particles in interplanetary space—I. The zodiacal dust cloud

The problem of electromagnetic perturbations of charged dust particle orbits in interplanetary space has been re-examined in the light of our better understanding of the large scale spatial and temporal interplanetary plasma and field topology. Using both analytical and numerical solutions for particle propagation it was shown that: (1) stochastic variations induced by electromagnetic forces are unimportant for the zodiacal dust cloud except for the lowest masses, (2) systemetic variations in orbit inclinations are unimportant if orbital radii are larger than 10 a.u. This is due to the solar cycle variation in magnetic polarity which tends to cancel out systematic effects, (3) systematic variations in orbital parameters (inclination, longitude of ascending node, longitude of perihel) induced by electromagnetic forces inside 1 a.u. tend to shift the plane of symmetry of the zodiacal dust cloud somewhat towards the solar magnetic equatorial plane, (4) inside 0.3 a.u. there is a possibility that dust particles may enter a region of “magnetically resonant” orbits for some time. Changes in orbit parameters are then correspondingly enhanced, (5) the observed similarity of the plane of symmetry of zodiacal light with the solar equatorial plane may be the effect of the interaction of charged interplanetary dust particles with the interplanetary magnetic field. Numerical orbit calculation of dust particles show that one of the results of this interaction is the rotation of the orbit plane about the solar rotational axis.     

Secular Motion in a 2nd Degree and Order-Gravity Field with no Rotation

The motion of a particle about a non-rotating 2nd degree and order-gravity field is investigated. Averaging conditions are applied to the particle motion and a qualitative analysis which reveals the general character of motion in this system is given. It is shown that the orbit plane will either be stationary or precess about the body's axis of minimum or maximum moment of inertia. It is also shown that the secular equations for this system can be integrated in terms of trigonometric, hyperbolic or elliptic functions. The explicit solutions are derived in all cases of interest.This revised version was published online in October 2005 with corrections to the Cover Date.     

On the periodically evolving orbits in the singly averaged Hill problem

We continue to analyze the periodic solutions of the singly averaged Hill problem. We have numerically constructed the families of solutions that correspond to periodically evolving satellite orbits for arbitrary initial values of their eccentricities and inclinations to the plane of motion of the perturbing body. The solutions obtained are compared with the numerical solutions of the rigorous (nonaveraged) equations of the restricted circular three-body problem. In particular, we have constructed a periodically evolving orbit for which the well-known Lidov-Kozai mechanism manifests itself, just as in the doubly averaged problem.     

Tidal evolution of close-in exoplanets in co-orbital configurations

In this paper, we study the behavior of a pair of co-orbital planets, both orbiting a central star on the same plane and undergoing tidal interactions. Our goal is to investigate final orbital configurations of the planets, initially involved in the 1/1 mean-motion resonance (MMR), after long-lasting tidal evolution. The study is done in the form of purely numerical simulations of the exact equations of motions accounting for gravitational and tidal forces. The results obtained show that, at least for equal mass planets, the combined effects of the resonant and tidal interactions provoke the orbital instability of the system, often resulting in collision between the planets. We first discuss the case of two hot-super-Earth planets, whose orbital dynamics can be easily understood in the frame of our semi-analytical model of the 1/1 MMR. Systems consisting of two hot-Saturn planets are also briefly discussed.     

Analysis of the orbit of Cosmos 72 (1965-53B) near 15th-order resonance

Cosmos 72 (1965-53B) was launched on 16 April 1965 into a near-circular orbit with an average height of 570 km and inclination 56°. Over the years, the orbit has contracted slowly under the influence of air drag, and On 27 June 1972 passed through exact 15th-order resonance, when successive equator crossings are 24° apart in longitude and the ground track repeats after 15 rev. The orbit has been determined at seven epochs between April 1972 and February 1973, using the RAE orbit refinement program PROP, with 544 optical and radar observations: the average orbital accuracy is about 50 m in height and 0.0008° in inclination.For Cosmos 72 the change in inclination at 15th-order resonance, due to perturbations by 15th-order harmonics in the geopotential, is greater than for any satellite previously analysed— nearly 0.07°—and analysis of the change, using the seven PROP orbits and 45 U.S. Navy orbits, yields equations accurate to 4 per cent for the geopotential coefficients of order 15 and odd degree (15, 17, 19 …). A similar analysis of the variation in eccentricity gives less accurate equations for coefficients of order 15 and even degree (16, 18 …). The variations in right ascension of the node and argument of perigee have also been analysed.     

Error analyses of resonant orbits for geodesy

An error analysis of resonant orbits for geodesy indicates that attempts to use resonance to recover high order geopotential coefficients may be seriously hampered by errors in the geopotential. This effect, plus the very high correlations (up to .99) of the resonant coefficients with each other and the orbital period in single satellite solutions, makesindividual resonant orbits of limited value for geodesy. Multiple-satellite, single-plane solutions are only a slight improvement over the single satellite case. Accurate determination of high order coefficients from low altitude resonant satellites requires multiple orbit planes and small drift-periods to reduce correlations and effects of errors of non-resonant geopotential terms. Also, the effects of gravity model errors on low-altitude resonant satellites make the use of tracking arcs exceeding two to three weeks of doubtful validity. Because high-altitude resonant orbits are less affected by non-resonant terms in the geopotential, much longer tracking arcs can be used for them.